Returns and risk
Measuring returns
Total shareholder return, and why the arithmetic and geometric means tell different stories.
Learning objectives
- Calculate holding-period and total shareholder returns, including dividends
- Calculate arithmetic and geometric mean returns from a price series
- Explain which average describes compound growth and annualise a monthly return
Every investment discussion starts with a return: "the shares are up 12% this year", "the fund averaged 9% a year". Those two statements can hide very different outcomes. A return that leaves out dividends understates what shareholders earned, and an average of yearly returns can be positive while the investor lost money.
This lesson builds the measures analysts use: holding-period return, total shareholder return (TSR) and the two mean returns. Bustan Foods Co. is a fictional listed food producer, and its share prices, the market index and all other figures are invented for teaching. Examples are educational and illustrative, not investment advice.
Total return = (P₁ + D − P₀) ÷ P₀ = price return + dividend yield
- P₀
- price at the start of the period
- P₁
- price at the end of the period
- D
- dividends received during the period (per share)
Arithmetic mean = (r₁ + r₂ + … + rₙ) ÷ n · Geometric mean = [(1 + r₁) × (1 + r₂) × … × (1 + rₙ)]^(1/n) − 1
- rₜ
- the return in period t (e.g. a monthly total return)
- n
- the number of periods
Annualising. A monthly geometric mean compounds to an annual figure: (1 + monthly)^12 − 1. Over exactly twelve months this equals the 12-month total return. Multiplying a monthly mean by 12 ignores compounding, and multiplying an arithmetic monthly mean by 12 stacks the two errors.
How an analyst reports a track record. Suppose you are summarising Bustan Foods Co.'s last three years for an investment committee. Step by step: (1) calculate each month's total return, including any dividend paid that month; (2) chain them — multiply the growth factors (1 + r) — to get the cumulative return; (3) convert that into an annual rate with the power 1/3; (4) state the price return alongside it, so readers can see how much came from dividends.
On the fictional data, the 36 monthly total returns compound to 29.8%, or 9.1% a year. The price alone moved 19.2%; the gap is the contribution of dividends and their reinvestment. Had you multiplied the arithmetic monthly mean by 12 you would have reported 10.7% a year — a figure no investor actually earned.
Assumptions to state. Chaining monthly total returns assumes each dividend is reinvested in the share when it is paid; an investor who took the cash out earned a slightly different result. Returns here are before costs and taxes. And a past return describes what happened over one particular window — change the start or end month and the figure can change materially, which is why the next lesson adds a measure of how bumpy the ride was.
Calculate
You bought a share at SAR 36.00, received dividends of SAR 1.10 during the year, and the share now trades at SAR 34.20. What is your one-year total shareholder return? (Percent, two decimals.)
Calculate
A portfolio returned +25%, -20%, +15% in three consecutive years. What was its compound (geometric) annual return? (Percent, two decimals.)
Check your understanding
A fund returned +60% in year 1 and -40% in year 2. Its factsheet reports an "average annual return of 10%". What happened to SAR 100 invested at the start?
Challenge: Bustan's 12-month returns
The sheet holds Bustan Foods Co.'s month-end share prices for months 24–36 (fictional data) and the dividend paid in month 30.
- In D3:D14, calculate each month's total return, including any dividend.
- In E3:E14, the growth factor 1 + return.
- In H2:H5, the arithmetic and geometric mean monthly returns, the 12-month total shareholder return (chain the growth factors, so the dividend is reinvested), and the 12-month price return.
Rows 7–8 then compare the two annualised means.
| A | B | C | D | E | F | G | H | |
|---|---|---|---|---|---|---|---|---|
| 1 | Month | Price (SAR) | Dividend (SAR) | Total return | Growth factor | Last 12 months | Result | |
| 2 | 24 | 25.90 | 0.00 | Arithmetic mean (monthly) | ||||
| 3 | 25 | 25.43 | 0.00 | Geometric mean (monthly) | ||||
| 4 | 26 | 25.28 | 0.00 | 12-month total shareholder return | ||||
| 5 | 27 | 25.25 | 0.00 | 12-month price return | ||||
| 6 | 28 | 23.81 | 0.00 | |||||
| 7 | 29 | 25.10 | 0.00 | Arithmetic mean × 12 (overstates) | 0.00% | |||
| 8 | 30 | 28.62 | 0.80 | Geometric mean, annualised | 0.00% | |||
| 9 | 31 | 29.63 | 0.00 | |||||
| 10 | 32 | 31.86 | 0.00 | |||||
| 11 | 33 | 29.00 | 0.00 | |||||
| 12 | 34 | 27.41 | 0.00 | |||||
| 13 | 35 | 26.04 | 0.00 | |||||
| 14 | 36 | 28.60 | 0.00 |
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